Transverse nonlinear instability for two-dimensional dispersive models

 Access to full text

 Full (PDF)

1. [1] C Alexander J., Pego R.L., Sachs R.L., On the transverse instability of solitary waves in the Kadomtsev–Petviashvili equation, Phys. Lett. A226 (1997) 187-192. Zbl0962.35505MR1435907
2. [2] Benjamin T., The stability of solitary waves, Proc. London Math. Soc. (3)328 (1972) 153-183. MR338584
3. [3] K. Blyuss, T. Bridges, G. Derks, Transverse instability and its long-term development for solitary waves of the $(2+1)$-Boussinesq equation, Preprint, 2002. 
4. [4] Bona J., Sachs R., Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys.118 (1988) 15-29. Zbl0654.35018MR954673
5. [5] Burq N., Gérard P., Tzvetkov N., Two singular dynamics of the nonlinear Schrödinger equation on a plane domain, Geom. Funct. Anal.13 (2003) 1-19. Zbl1044.35084MR1978490
6. [6] Cazenave T., Lions P.L., Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys.85 (1982) 549-561. Zbl0513.35007MR677997
7. [7] Coppel W.A., Dichotomies in Stability Theory, Lecture Notes in Mathematics, vol. 629, Springer-Verlag, Berlin, 1978. Zbl0376.34001MR481196
8. [8] Friedlander S., Strauss W., Vishik M., Nonlinear instability in an ideal fluid, Ann. Inst. H. Poincaré14 (1997) 187-209. Zbl0874.76026MR1441392
9. [9] Friedlander S., Vishik M., Nonlinear instability in two-dimensional ideal fluids: The case of a dominant eigenvalue, Comm. Math. Phys.243 (2003) 261-273. Zbl1043.76025MR2021907
10. [10] Grenier E., On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math.53 (2000) 1067-1091. Zbl1048.35081MR1761409
11. [11] Guo Y., Strauss W.A., Instability of periodic BGK equilibria, Comm. Pure Appl. Math.48 (1995) 861-894. Zbl0840.45012MR1361017
12. [12] Henry D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin, 1981. Zbl0456.35001MR610244
13. [13] A. Ionescu, C. Kenig, Local and global well-posedness of periodic KP-I equations, Preprint, 2005. MR2150876
14. [14] Janssen P., Rasmussen J., Nonlinear evolution of the transverse instability of plane envelope solitons, Phys. Fluids26 (1983) 1279-1287. Zbl0521.76024MR717346
15. [15] Kadomtsev B.B., Petviashvili V.I., On the stability of solitary waves in weakly dispersive media, Soviet Phys. Dokl.15 (1970) 539-541. Zbl0217.25004
16. [16] Kato T., Perturbation Theory for Linear Operators, Classics in Mathematics, Reprint of the 1980 edition, Springer-Verlag, Berlin, 1995. Zbl0836.47009MR1335452
17. [17] Kenig C., Ponce G., Vega L., Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Comm. Pure Appl. Math.46 (1993) 527-629. Zbl0808.35128MR1211741
18. [18] Koch H., Tzvetkov N., On finite energy solutions for the KP-I equation, Math. Z.256 (2008) 55-68. Zbl05218251MR2350033
19. [19] Liu Y., Strong instability of solitary wave solutions to a Kadomtsev–Petviashvili equation in three dimensions, J. Differential Equations (2002) 153-170. Zbl1061.35115MR1890602
20. [20] Merle F., Vega L., $L^2$ stability of solitons for KdV equation, Int. Math. Res. Notices13 (2003) 735-753. Zbl1022.35061MR1949297
21. [21] Pego R., Weinstein M., Eigenvalues, and instabilities of solitary waves, Philos. Trans. Roy. Soc. London A340 (1992) 47-97. Zbl0776.35065MR1177566
22. [22] Saut J.-C., Remarks on the generalized Kadomtsev–Petviashvili equations, Indiana Univ. Math. J.42 (1993) 1011-1026. Zbl0814.35119MR1254130
23. [23] Takaoka H., Tzvetkov N., On 2D nonlinear Schrödinger equations with data on $R\times T$, J. Funct. Anal.182 (2001) 427-442. Zbl0976.35085MR1828800
24. [24] Titchmarch E.C., Eigenfunction Expansions Associated to Second Order Differential Equations, Clarendon Press, Oxford, 1946. Zbl0061.13505MR19765
25. [25] Weinstein M., Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal.16 (1985) 472-491. Zbl0583.35028MR783974
26. [26] Zakharov V.E., Instability and nonlinear oscillations of solitons, JETP Lett.22 (1975) 172-173. 

1. Martin Hadac, Sebastian Herr, Herbert Koch, Well-posedness and scattering for the KP-II equation in a critical space